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| This article needs updating in light of changes to the solver in March 2010, particularly the representation of chains Chaining strategies like X-Cycles use links that connect cells on the board. It is possible to expand on the idea of a "cell" to something a bit larger, namely a "group" of cells. I prefer the word "node" - which in 90% of cases will be a single cell - but can be two or three cells in the same unit. You might wonder how we can use more than one cell and think of it as a node between two links, but there is some cool logic here. We must go all the way back to Pointing Pairs and Pointing Triples. They attack cells along the row or column on which they are aligned. They also must be in the same box to be a coherent unit. Our “grouped” cells are just Pointing Pairs/Triples and we’re going to use them as part of a chain or Nice Loop. |
| Clearing the clutter on an example board, in Figure 1, we have a spread of candidate 4s. All the lettered cells are also candidate 4. There is a continuous Nice Loop starting with A. B-C is a weak link, and so is D-E. The interesting part is the set of cells {X,Y,Z}. It does not matter which of X, Y, or Z (if any) is the solution; any of them will eliminate A and E. Likewise, if E is true, then all of XYZ are gone – and A is true. We can think of {X,Y,Z} as a single node for the purposes of our logic. This promotes the links from A and E to strong links, and the notation for this part of a loop is: 4[F8]=4[D7|D8|D9]=4[D3]= E X or Y or Z A |
 Figure 1: Grouped X-Cycle |
| The important characteristic is that the cells are all in the same box. One end of the chain (in this case, A) is pointed to by the node cells; the other (in this case, E) is usually within the same box as the node. |
| Nice Loop Rule 1 Check the article on X-Cycles for a review of Nice Loop Rule 1 in a cycle that does not contain Grouped Nodes. Figure 2 invokes Nice Loop Rule 1. The output from the solver says: X-CYCLE on 3 (Grouped Continuous Alternating Nice Loop, length 6): 3[A1]-3[E1]=3[E9]-3[H9]=3[G7|H7]-3[A7]=3[A1]- The Grouped node is {G7,H7} and it points up column 7 to A7 (a weak link since there is another 3 at B7. Five 3s can be eliminated in one fell swoop since all 3s that are on units with links of weak inference can go. |
 Figure 2: Grouped 3-Cycle: Load Example or : From the Start |
| This is a much more complex example, but it does show how powerful the strategy is in tackling a bottleneck on the board. The Nice Loop contains two Grouped nodes at {G3,J3} and {G8,G9}. The first is aligned on column 3 and points to D3. The second points along the row to G2. All other 8s in any unit shared by any of the Weak links can be removed (Nice Loop Rule 1). X-CYCLE: 8 taken off E8 (link: D8 / H8) X-CYCLE: 8 taken off E3 (link: G3 / D3) X-CYCLE: 8 taken off F3 (link: G3 / D3) X-CYCLE on 8 (Grouped Continuous Alternating Nice Loop, length 6): 8[D3]=8[D8]-8[H8]=8[G8|G9]-8[G2]=8[G3|J3]-8[D3]= |
 Figure 3: Grouped 8-Cycle: Load Example or : From the Start |
| Nice Loop Rule 2 Check the article on X-Cycles for a review of Nice Loop Rule 2 in a cycle that does not contain Grouped Nodes. In Figure 4 we have a an X-Cycle on 1 (a 1-Cycle) with a group of two {A3,B3} usefully working together as a node. This allows us to create a chain linking the coloured cells on this diagram. From Rule 2 we can deduce that H9 must be a 1. X-CYCLE on 1 (Grouped Discontinuous Alternating Nice Loop, length 5): 1[H9]=1[C9]-1[C2]=1[A3|B3]-1[H3]=1[H9]= - Discontinuity is two strong links joined at H9, all other candidates (9) can be removed from that cell |
 Figure 4: Grouped 1-Cycle: Load Example or : From the Start |
In my second example in Figure 5 the chain is a little longer and bit more compact, but the grouped node {G4,G5} on 6 makes the chain possible. X-CYCLE on 6 (Grouped Discontinuous Alternating Nice Loop, length 7): 6[C6]=6[C5]-6[J5]=6[G4|G5]-6[G8]=6[E8]-6[E6]=6[C6]= - Discontinuity is two strong links joined at C6, all other candidates (8) can be removed from that cell Rule 2 examples are probably the least likely of the three rules to be found. Most grouped X-Cycles will be two weak links and Rule 3, next. |
 Figure 5: Grouped 6-Cycle: Load Example or : From the Start |
| Nice Loop Rule 3 I couldn't resist this example as it has two grouped nodes - {B1,C1} and {E7,F7}. Don't be confused about {F9} and {G9}, they are just adjacent yellow cells that have a weak link on 2. You can see plenty of other 2s in column 9, but a grouped node has to be in the same box. Rule 3 tells us that two weak links joined at B7 then 2 can be removed from that cell. X-CYCLE on 2 (Grouped Discontinuous Alternating Nice Loop, length 6): 2[B7]-2[B3]=2[B1|C1]-2[G1]=2[G9]-2[F9]=2[E7|F7]-2[B7]- |
 Figure 6: Grouped 2-Cycle: Load Example or : From the Start |
| If you want to continue reading about this strategy family, the next chapter is entitled Alternating Inference Chains. |
